High-performance superconducting quantum processors via laser annealing of transmon qubits

Scaling the number of qubits while maintaining high-fidelity quantum gates remains a key challenge for quantum computing. Presently, superconducting quantum processors with >50 qubits are actively available. For these systems, fixed-frequency transmons are attractive because of their long coherence and noise immunity. However, scaling fixed-frequency architectures proves challenging because of precise relative frequency requirements. Here, we use laser annealing to selectively tune transmon qubits into desired frequency patterns. Statistics over hundreds of annealed qubits demonstrate an empirical tuning precision of 18.5 MHz, with no measurable impact on qubit coherence. We quantify gate error statistics on a tuned 65-qubit processor, with median two-qubit gate fidelity of 98.7%. Baseline tuning statistics yield a frequency-equivalent resistance precision of 4.7 MHz, sufficient for high-yield scaling beyond 103 qubit levels. Moving forward, we anticipate selective laser annealing to play a central role in scaling fixed-frequency architectures.


INTRODUCTION
Recent technological advances have enabled rapid scaling of both the physical number of qubits and computational capabilities of quantum computers (1)(2)(3)(4)(5). The distinction between classical and quantum computing arises from the exponentially larger computa tional subspace available to qubits (quantum bits), which may be set to nonclassical superpositions and entangled states. Existing multi qubit systems built on superconducting circuit quantum electro dynamics (cQED) architectures (6,7) have been used in applications ranging from early implementations of Shor's factoring algorithm (8) to quantum chemistry simulations (9)(10)(11) and accelerated feature mapping in machine learning (12)(13)(14). Presently, solidstate cQED based processors have substantially increased in scale (2,15), with dozens of physical qubits demonstrated on a single quantum chip. As gate fidelities improve and eventually reach thresholds required for fault tolerance, quantum advantage will be exploited to simulate complex molecular dynamics and implement quantum algorithms on practical scales (9,(16)(17)(18). To track the continual progression of quantum processing power, the quantum volume (QV) metric is used as an overall measure of the computational space available for a given processor (5).
Among the major classes of superconducting qubits, fixed frequency transmons (3,7) operating in the E J >> E C regime are attractive for their low charge dispersion, yielding relatively noise immune qubits with coherence times (T 1 , T 2 ) exceeding 100 s (19). Transmon qubits are amenable to highfidelity (>99.9%) singlequbit gate operations (3), and twoqubit entangling crossresonance (CR) gates (20) are realized via static qubitqubit coupling activated by an allmicrowave drive scheme (3,21). A key requirement to enable highfidelity CR gates involves the selective addressability of fixed frequency transmons, as well as precisely engineering their compu tational |0〉 → |1〉 transitions (f 01 ) for optimal twoqubit interaction (3). For example, suboptimal f 01 separation between neighboring qubits reduces ZX coupling strength, while higherorder static ZZ interactions cause accumulation of twoqubit errors and "spectator" error propagation across the lattice (22). The most likely frequency collisions and corresponding tolerance bounds resulting from frequency crowding of lattice transmons have been quantitatively enumerated in (23).
The principal challenge for scaling fixedfrequency architectures is mitigating errors arising from lattice frequency collisions. Typical fabrication tolerances for transmon frequencies range from 1 to 2%, with uncertainties dominated by the 2 to 4% variation in tunnel junction resistance R n (23,24). Thermal annealing methods to adjust and stabilize postfabrication R n (and correspondingly, transmon frequencies f 01 ) have been explored previously in (25)(26)(27)(28). More recently, the LASIQ (Laser Annealing of Stochastically Im paired Qubits) technique was introduced to increase collisionfree yield of transmon lattices by selectively trimming (i.e., tuning) indi vidual qubit frequencies via laser thermal annealing (23). The LASIQ process sets R n with high precision, and f 01 could be predicted from R n according to a powerlaw relationship resulting from the AmbegaokarBaratoff relations and transmon theory (7,29). An empirical f 01 (R n ) scatter of  f ≃ 14 MHz limited the ability to predict f 01 from R n , resulting in a posttune frequency precision of equiva lently ~14 MHz (23).
Here, we demonstrate LASIQ as a scalable process tool used to reduce twoqubit gate errors by systematically trimming lattice transmon frequencies to desired patterns. We show laser tuning re sults on statistical aggregates of >300 tuned qubits and demonstrate LASIQ baseline frequencyequivalent resistance tuning precision of 4.7 MHz from empirical f 01 (R n ) correlations, reaching this precision with 89.5% success rate. On the basis of cryogenic f 01 measurements from our lasertuned processors, we empirically find a frequency assignment precision of 18.5 MHz, which, in addition to the LASIQ tuning precision, includes all deviations arising from precooldown steps including posttuned chip cleaning and bonding processes.

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Our results indicate that the precision of trimming f 01 is dominated by the residual f 01 (R n ) scatter of untuned qubits ( f = 18.1 MHz) and is not limited by the LASIQ trimming process itself.
In addition to scaling the number of tuned qubits, we measure functional parameters of multiqubit chips (coherence and twoqubit gate fidelity) to ensure high processor performance. We assess the impact of LASIQ tuning on qubit coherence using a collection of composite (partially tuned) processors, showing aggregate mean T 1 and T 2 times of 79 ± 16 s and 69 ± 26 s, respectively, with no statistically significant variation between the tuned and untuned groups. Our LASIQ process has been broadly used for precise fre quency control of postfabricated 27qubit Falcon processors, in cluding the recent QV128 ibmq_montreal system (5,30). We demonstrate LASIQ scaling capabilities by tuning a 65qubit Hummingbird processor (cloudaccessible as ibmq_manhattan), with a median twoqubit gate fidelity of 98.7% based on randomized benchmarking (3,31). As a scalable frequency trimming tool for fixedfrequency transmon architectures, we envision the LASIQ process to be widely implemented in future generations of super conducting quantum systems.

Tuning a 27-qubit Falcon processor
As a practical demonstration of frequency trimming, a 27qubit Falcon processor is tuned to predicted frequency targets using the LASIQ setup shown in Fig. 1A (described in Materials and Methods). The Falcon chip series are based on a heavyhexagonal lattice, which contains the distance3 hybrid surface and BaconShor code for error correction (32). For this demonstration, all measurements were performed at ambient conditions, with resulting tuned fre quencies estimated from empirical f 01 (R n ) correlations. The tuned lattice is depicted in Fig. 1B, with a color heatmap indicating the postLASIQ frequency predictions. Nearestneighbor (NN) qubit frequency spacing (f NN ) can be visually seen to be separated by 50 MHz ≤ |f NN | ≤ 250 MHz, placing qubits comfortably within the straddling regime for highZX interaction (33). NN collisions have been avoided with twice the collision tolerance (2 c , see the Supplementary Materials) from bounds described in (23) to protect against twoqubit state hybridization and improve chip yield. Before tuning, highrisk pairs within 2 c collision bounds have edge borders highlighted as indicated in the lattice graph and are resolved after the LASIQ process is complete. On the basis of Monte Carlo models with a conservative posttune spread estimate of  f = 20 MHz (see the next section), we demonstrate substantial increase in the yield rate from 3.4 to 51% for achieving zero NN collisions, corre sponding to 15× yield improvement after LASIQ tuning (see the Supplementary Materials). The target qubit f 01 patterns corresponding to Fig. 1B are shown in Fig. 1C (bottom), where initial f 01 (red) are progressively tuned until they reach predefined and distinct fre quency set points (blue).
In addition to NN collision avoidance, all qubits have been tuned to targets at or below f purc = 5.2 GHz, which, in our Falcon design, is the cutoff for maintaining good Purcell protection and avoiding radiative qubit relaxation (34). Because of the monotonic increase of junction resistance (R n ) intrinsic to the laser anneal process, qubit frequencies are "trimmed" (i.e., reduced) to desired f 01 values. The LASIQ process is engineered to proceed until R n for all junction reside within 0.3% of the target R T [corresponding to ~10 MHz for typical f 01 (R n ) correlation coefficients], although a nominal LASIQ approach will typically outperform this upper precision bound (see the next section). The middle panel of Fig. 1C shows desired target shifts (purple diamonds) superimposed on final predicted frequency tuning amplitudes (green bars), indicating excellent agreement with desired target values. The root mean square (RMS) R n deviation from R T for this processor is 0.16%, corresponding to a frequencyequivalent resistance tuning precision of 4.8 MHz [using nominal f 01 (R n ) powerlaw coefficients for this sample] and is consistent with LASIQ baseline tuning precision as described in Fig. 2. As seen from the middle panel, target f 01 shifts range from 75 MHz (qubit 8) to 375 MHz (qubit 21) in Fig. 1C, corresponding to resistance shifts up to R n = 14.2%. On the basis of separate calibration measurements over standard junction arrays, resistance tuning peaks at ~14%, and therefore, tuning plans for each processor are designed to be con strained within attainable targets. Upon completion of the LASIQ process, a typical quantum processor is cooled and screened for coherence and single/twoqubit gate fidelity, and assessments of QV are performed (5). Twoqubit gate error statistics of a tuned and operational 65qubit Hummingbird processor are shown in the "Qubit coherence and gate fidelity" section.

LASIQ tuning precision
In prior work, tuning precision estimates were limited by the im precision in predicting f 01 from room temperature R n measurements (23). Thus, only an upper bound of tuning precision could be deter mined ( f ≃ 14 MHz). Presently, we address the limits of LASIQ tuning precision as limited by the process itself, rather than our ability to predict cryogenic f 01 (R n ). We analyze a large sample of 390 tuned qubits, 349 of which successfully tuned to target (defined in the previous section as achieving junction resistance within 0.3% of R T ), yielding an aggregate tuning success rate of 89.5% for this experiment. The initial and final distributions of successfully tuned qubits are shown in Fig. 2. We note here that a full range of tuning (R up to ~14% with respect to initial junction R n ) was performed for this experiment, with anticipated failure rates being weighted toward tuning extremes (R > 14% increases undershoot risk, while low tuning R < 1% results in increased overshoot risk). These extremes are readily avoided during f 01 tuning plan assignment. A detailed analysis of tuning statistics and success rates has been per formed for all qubits (see the Supplementary Materials).
To understand the consequences of the incremental approach to R T , aggregate tuning statistics of the 349 successfully tuned qubits are depicted in Fig. 2A. R n of the initial qubits (gray histogram) are tuned to a tight tolerance about the targets (orange), as displayed on an R T normalized scale. A Gaussian fit approximates the initial re sistance distribution, yielding a mean fractional tuning distance of 6.7% (with respect to R T ). The final distribution is magnified in Fig. 2B with a superimposed lognormal curve fit, a characteristic distribution for fractional growth processes consistent with incre mental anneals during the LASIQ process. Averaging over the entire distribution yields an RMS error deviation  R = 0.17% from R T , cor responding to  f = ∂f 01 /∂R n •  R = 4.7 MHz as determined by empir ical f 01 (R n ) correlations from our 65qubit Hummingbird processor (Fig. 3). Our determination of LASIQ resistance precision shows that the fundamental tuning performance is well below the upper bound of ~14 MHz determined in (23), where it was also noted that this bound was dominated by residual scatter in the prediction of cryogenic f 01 from room temperature R n . On the basis of Monte Carlo simulations, a precision of ≤6 MHz is required for highyield scal ing for quantum processors beyond 10 3 qubits (23). Therefore, our baseline frequencyequivalent resistance precision of  f < 5 MHz demonstrates LASIQ as a viable postfabrication trimming process for highyield scaling of fixedfrequency transmon processors.
In addition to the residual f 01 (R n ) prediction scatter, a number of preparatory cleaning, bonding, and processor mounting steps are undertaken for our quantum processors in the precooldown stage. A natural question therefore arises as to the practical precision achieved for frequency assignment with the inclusion of these pro cesses. Figure 3A shows cryogenic f 01 plotted against postLASIQ measurements of R n on a 65qubit Hummingbird processor, which comprised 49 tuned qubits and 16 untuned qubits. After tuning, the processor underwent plasma cleaning and flipchip bumpbonding to an interposer layer before mounting in a dilution refrigerator for cooldown and screening (see Materials and Methods). The entire process occurred within a 24hour span to minimize the impact of aging and drift on the junction resistances (and therefore qubit fre quencies). A powerlaw fit (dashed curve) conforms well to both the tuned and untuned qubits, indicating that no appreciable relative shifts due to LASIQ tuning have occurred, and that the same f 01 (R n ) prediction may be adequately used in both cases. We note the slight deviation of the powerlaw fit exponent (−0.55) from the nominal onehalf expected from the AmbegaokarBaratoff relations and transmon theory (7,29), which is attributed to nonidealities in pre dicting f 01 from room temperature R n . Nevertheless, this deviation to first order does not affect the relative frequency spacing between qubits and, therefore, does not affect the precision to which we as sign qubit frequency spacing for collision avoidance. The effective f 01 assignment precision may be determined by the residuals of the powerlaw fit as shown in Fig. 3A (inset), outlined by a Gaussian distribution with  f = 18.6 MHz spread, corresponding to 0.38% of the mean qubit frequency (4.84 GHz), and defines the practical f 01 assignment precision for this chip. Figure 3B shows a similar analysis performed over a statistical sampling of 241 qubits from seven Falcon and two Hummingbird tuned processors. Each processor sample was fit to an individual f 01 (R n ) curve, and residuals are aggregated in the upper histogram (dark green), with 1 f spread of 18.5 MHz, consistent with the singleprocessor sample observed in Fig. 3A. Results from control (untuned) qubits are shown in the bottom panel histogram (gray), which are f 01 (R n ) residuals extracted from 117 untuned qubits on composite (partially tuned) processors, yielding 1 f spread of 18.1 MHz. We may therefore conclude that the f 01 assignment im precision resulting from our LASIQ process is a negligible contrib utor to the overall frequency spread of the qubits. We note that although our residual value for both tuned and untuned samples is larger than the ~14 MHz demonstrated in (23), our Falcon and Hummingbird processors undergo a greater number of preparatory steps before cooldown, and a certain amount of deviation is antici pated. The prediction imprecision of f 01 (R n ) remains the dominant contributor to overall spread, with relatively smaller contributions from posttune drifts and precooldown processes. Therefore, room remains for improving frequency predictions to reach singleMHz levels dictated by the ~5 MHz LASIQ baseline frequencyequivalent tuning precision as shown in Fig. 2B (details in Discussion).

Qubit coherence and gate fidelity
Maintaining high qubit coherence is an essential component of highfidelity single and twoqubit gates, in addition to precise frequency control. To empirically determine the impact of laser tuning on qubit coherence, a composite (partially tuned) set of four Hummingbird processors were cooled and coherence was assessed. The composite chips allow direct comparison of tuned and untuned qubits drawn from the same initial population. Of a total 221 mea sured qubits from the four composite processors, 162 were LASIQ tuned and 59 were left untuned (73% fractional tuning rate). The statistical aggregates of tuned and untuned (T 1 , red) and dephasing (T 2 , blue) times are shown in Fig. 4. For tuned (untuned) qubits, 〈T 1 〉 = 80 ± 16 s (76 ± 15 s) and 〈T 2 〉 = 68 ± 25 s (70 ± 26 s). Aggregate (including both tuned and untuned qubits) relaxation and dephasing times are 79 ± 16 s and 69 ± 26 s, respectively. Box plots (interquartile box range, with 10 to 90% whiskers) shown in Fig. 4A demonstrate that, within statistical error, the LASIQ process introduces no variation in qubit coherence. Figure 4B offers a de tailed comparison of the tuned and untuned coherence distributions on a quantilequantile plot. Good correspondence is observed with respect to linear unity slope, indicating close agreement between the distributions. Visual indicators of the mean 〈T 1 〉 and 〈T 2 〉 and 1 bounds are shown in the shaded ovals. Together, these compar isons of statistical distributions demonstrate the negligible effect of the LASIQ process on qubit coherence.
As a practical demonstration of LASIQ tuning capabilities, a 65qubit Hummingbird processor is lasertuned and operationally cloudaccessible as ibmq_manhattan. In a similar fashion to that shown in Fig. 1B, the LASIQ tuning plan is generated by ensuring avoidance of NN level degeneracies while maintaining level separa tion within the straddling regime (33). After LASIQ, the 65qubit processor was cooled and qubit frequencies were measured, with density of frequency detuning between twoqubit gate pairs shown in Fig. 5A (orange, 10 MHz bin width), along with the initial twoqubit detuning (blue, 30 MHz bins). As tuned, our processor consists of 72 operational twoqubit CR gates, corresponding to a 100% yield of working twoqubit gates, with gate durations ranging from 250 to 600 ns. Collision boundaries [2 c bounds as derived from (23,33); see the Supplementary Materials] for NN degeneracies are shown in the background, consistent with those used in Fig. 1A, indicating that ~20% of gates are within "highrisk" collision zones. Monte Carlo yield modeling of the untuned (asfabricated) 65qubit Hummingbird chip indicates an average of 12 NN collisions (assuming nominal  Fig. 5B, indicating good separation near nulldetuning (type 1 collision) while maintaining tight ZZ distri bution with median 69 kHz (±23.2 kHz). We note that transmon transmon coupling suffers from staticZZ (i.e., "alwayson") error; however, within the f j,k <  regime (anharmonicity  ≃ −330 MHz; see Materials and Methods), tailoring the ZZ distribution to present magnitudes (~70 kHz) with collisionfree detuning is sufficient to yield gate errors approaching 10 −2 for typical twoqubit CR gate du rations of ~400 ns, indicating that our twoqubit gate fidelities are near levels set by coherence (22). Twoqubit gate errors as a function of qubitpair detuning are displayed in Fig. 5C, with their associated distribution shown in the adjacent probability density map (right). Notably, our detuning dis tribution shows more gates with positive controltarget detuning, as expected given the greater ZX interaction in the positive detuning region. A CR gate error model is also depicted in the figure (see Materials and Methods), where the shaded background (gray) indi cates lowfidelity regions consistent with known frequency collisions described in (23). As part of our model, an optimizer routine incor porates a standard CR echo sequence with optional target rotary pulsing to determine usable twoqubit detuning regions within the straddling regime. For a gate error of 1%, a total usable frequency range of 380 MHz is available (optimized using J = 1.75 MHz in the depicted model), which reduces to 350 MHz (130 MHz) for error targets of 0.5% (0.1%). We note that our depicted model does not incorporate the impacts of classical crosstalk and coherence, the latter of which limits gate error in the desirable detuning regions. On the basis of the preLASIQ detuning distribution in Fig. 5A and the CR gate error model in Fig. 5C, we predict an initial mean gate error of 5.7%, which is improved to 1.4% after LASIQ tuning and demonstrates the efficacy of LASIQ in improving twoqubit gate fidelity (see the Supplementary Materials). Last, we note that although our collision bounds and unitary gate model have similar qualita tive outcomes, further work is required to determine exact collision constraints and identify highfidelity detuning regimes as lattice sizes are progressively increased.

DISCUSSION
Significant yield improvement and high twoqubit gate fidelities for both Falcon and Hummingbird processors demonstrate LASIQ as an effective postfabrication frequency trimming technique for multiqubit processors based on fixedfrequency transmon architec tures. Selective laser annealing offers a compelling and scalable solution to the problem of frequency crowding, with LASIQ being readily adaptable to the scaling of qubits on progressively larger quantum processors. On the basis of Monte Carlo models (23), the 4.7MHz frequencyequivalent resistance tuning precision of LASIQ allows highyield scaling up to and beyond the 1000qubit level. At present, cryogenic f 01 measurements indicate a practical precision 0 5 0 100   of 18.5 MHz; however, statistical analysis of cryogenictoambient thermal cycling in a dilution refrigerator yields a recool stability of 5.7 MHz for our multiqubit processors, which may be leveraged to obtain f 01 (R n ) predictions with similar accuracy and approach the baseline frequency assignment precision allowed by LASIQ. Last, we note that despite our present emphasis on NN collisions, errors arising from nextNN qubits (e.g., spectator collisions) are a rising concern with lattice scaling, and future work will incorporate tuning plans to minimize these errors, as well as lattice frequency pattern optimization for yield maximization.

Laser annealing for transmon frequency allocation
The LASIQ frequency tuning system is depicted in Fig. 1A, whereby a diodepumped solidstate laser (532 nm) is actively aligned and focused on a multiqubit quantum processor to selectively anneal individual transmon qubits (23). A timed shutter precisely controls the anneal duration and monotonically increments the junction re sistance over multiple exposures, with intermediate resistance mea surements to aid the adaptive approach to target resistances (R T ). Diffractive beam shaping is optionally available to aid in the uniform thermal loading of the junction (36). The LASIQ process is fully automated and tunes an entire chip to completion, which occurs when each junction is annealed to within 0.3% tolerance around R T [corresponding to ~10 MHz for typical f 01 (R n ) correlations].

Preparation and screening of multiqubit processors
The Josephson junctions are fabricated via standard electron beam lithographic patterning followed by twoangle shadow evaporation with intermediate oxidation for Al/AlO 2 /Al junction formation (37). Typical lateral dimensions of the junction are ~100 nm, which yield asfabricated frequency deviations near 2%, corresponding to ~100 MHz of qubit frequency spread. Transmon qubit designs follow those described in (5,23), with coupled qubits in a fixedfrequency lattice architecture. Nominal qubit anharmonicities are engineered near  ≃ −330 MHz. Following junction evaporation, prescreening of viable processor candidates is performed via room temperature resistance measurements of all Josephson junctions, with transmon qubit frequencies estimated through waferlevel f 01 (R n ) powerlaw fits from prior control experiments. Candidates are ranked on the basis of suitability for LASIQ tuning into collisionfree tuning plans and within the Purcell filter bandwidth, all under the constraint of tuning range limits (typical maximum resistance tuning of ~14%). Following this preselection process, a frequency plan is generated and tested through Monte Carlo collision and yield modeling (23). Automated LASIQ tuning is performed as defined by the tuning plan for each transmon qubit to a junction resistance precision band of 0.3%, which yields an astuned RMS frequencyequivalent resistance precision of ~5 MHz (see the "LASIQ tuning precision"). After LASIQ, the multiqubit processors are plasmacleaned and flipchip bonded to an interposer layer for Purcell filtering and readout. The bonded processors are mounted into a dilution refrigerator for cryogenic screening, including coherence measurements and single and twoqubit gate error analysis. The postLASIQ cleaning, bonding, and mounting steps are coordinated to occur within a 24hour span to minimize junction aging and drifts, thereby main taining the qubits faithfully outside collision bounds of the intended tuning frequency plans.

Gate error model
Our gate error estimates are the result of timedomain simulations of the Schrodinger equation within the Duffing model of two coupled transmon qubits (33). CR pulses are modeled as "Gaussian square" in shape. For each set of parameters, CR gate amplitudes are tuned up in a twopulse echo for a fixed gate time by minimizing the Bloch vector (5). A rotary pulse is additionally applied to the target qubit during the CR pulses, the amplitude of which is optimized to minimize gate error, as described in (22). Using an adaptive RungeKutta solver, we calculate the full unitary evolution of the system resulting from the calibrated gate sequence. From this simulated unitary U, error can be estimated by E = 1 -[1/d × |Tr(U target U † )| 2 + 1]/(d + 1) (where d = 2 n and n = 2 for our twoqubit simulations) and is com pared to the results of twoqubit randomized benchmarking.